Integral instability

The method is a necessary but not sufficient condition for stability of a closedloop system with integral action If the index is negative, the system will be unstable for any controller settings (this is called integral instability ). If the... 

The RGA is useful for avoiding poor pairings. If the diagonal element in the RGA is negative, the system may show integral instability the same situation that we discussed in the use of the Niederlinski index. Very large values of the RGA indicate that the system can be quite sensitive to changes in the parameter values. 

Hybridomas may be unstable because the techniques used to create them are fairly crude and may lead to a loss of cellular and genetic integrity. Instability is characterized by cell death after a few divisions or change in specificity. It is vitally important to test the hybridomas repeatedly to ensure that loss of specificity or abil-... 

The ti eatment of the Jahn-Teller effect for more complicated cases is similar. The general conclusion is that the appearance of a linear term in the off-diagonal matrix elements H+- and H-+ leads always to an instability at the most symmetric configuration due to the fact that integrals of the type do not vanish there when the product < / > / has the same species as a nontotally symmetiic vibration (see Appendix E). If T is the species of the degenerate electronic wave functions, the species of will be that of T, ... 

In general, type zero are unsatisfaetory unless the open-loop gain K ean be raised, without instability, to a suffieiently high value to make 1/(1 + A p) aeeeptably small. Most eontrol systems are type one, the integrator either oeeurring naturally, or deliberately ineluded in the form of integral eontrol aetion, i.e. PI or PID. 

For models described by a set of ordinary differential equations there are a few modifications we may consider implementing that enhance the performance (robustness) of the Gauss-Newton method. The issues that one needs to address more carefully are (i) numerical instability during the integration of the state and sensitivity equations, (ii) ways to enlarge the region of convergence. 

It should be emphasized here that it is unnecessary to integrate the state equations for the entire data length for each value of u. Once the objective function becomes greater than S(k0)). a smaller value for p can be chosen. By this procedure, besides the savings in computation time, numerical instability is also avoided since the objective function often becomes large very quickly and integration is stopped well before computer overflow is threatened. 

Furthermore, the implementation of the Gauss-Newton method also incorporated the use of the pseudo-inverse method to avoid instabilities caused by the ill-conditioning of matrix A as discussed in Chapter 8. In reservoir simulation this may occur for example when a parameter zone is outside the drainage radius of a well and is therefore not observable from the well data. Most importantly, in order to realize substantial savings in computation time, the sequential computation of the sensitivity coefficients discussed in detail in Section 10.3.1 was implemented. Finally, the numerical integration procedure that was used was a fully implicit one to ensure stability and convergence over a wide range of parameter estimates. 

Here, Rc is a cutoff distance which is made necessary to prevent the potential to become too steep in certain points of the grid and generate instabilities in the numerical integration, and is generally between 0.2 and 0.3A. 

Disadvantages of the integral control mode are that it responds relatively slowly to an error signal and that it can initially allow a large deviation at the instant the error is produced. This can lead to system instability and cyclic operation. For this reason, the integral control mode is not normally used alone, but is combined with another control mode. 

---